**Analysis of the features of quantum frequency standards **

**given in the forms of maximum of output power **

**vs frequency curve of a single-frequency gas laser **

**and Lamb dip centre**

**Hen ryk Percak**

**Institute of Telecommunication and Acoustics, Technical University of Wroclaw, W ybrzeże **
**Wyspiańskiego 27, 50-370 Wroclaw, Poland.**

**The purpose of this paper is to give an original theoretical explanation of the differen**
**ces of fundamental features of quantum frequency standards which up till now have **
**been found out only experimentally. It has been proved, that the maximum of the out**
**put power vs the frequency curve of single-frequency gas laser (further called the maxi**
**mum of OPF curve) in the case of approximately Gaussian OPF curve is frequency **
**standard about two times better than in the case of approximately Lorentzian OPF **
**curve. It has been proved also, that Lamb dip centre is frequency standard about ten **
**times better than the maximum of approximately Gaussian OPF curve and about **
**twenty times better than the maximum of approximately Lorentzian OPF curve.**

**In gas lasers, in which total pressure of gas gain mixture is high (for example, **
**in C02-hr2-He laser, pressure of about 2000 Pa or higher, at typical temperature **
**of about 400 K [1]), Lamb dip does not occur. It is so, because homogeneous **
**broadening of vibration-rotation transitions, caused mostly by collision broa**
**dening, is then comparable with inhomogeneous broadening, caused by Doppler **
**effect. However, when the total pressure of gas gain mixture is low (for example, **
**in C02-N2-He laser, pressure of about 130 Pa or less, at typical temperature of **
**about 400 K [2]), Lamb dip is very sharply outlined, because then inhomoge**
**neous broadening predominates over the homogeneous one.**

**An emission line of gain medium can be described by the Lorentzian func**
**tion [3-5] in high total pressure range. The output power vs frequency curve **
**of single-frequency gas laser, further called OPF curve, resulting from such **
**kind of gain medium, can be approximately described by Lorentzian function **
**as well. In low total pressure range Lamb dip is also described by Lorenztian **
**function, and appears in the background of OPF curve, which is approximately **
**described by the Gaussian function [3-5] (see figure). The background curve **
**may be approximately described by the Gaussian function since due to Doppler **
**effect it is inhomogeneously broadened. Lamb dip appears in the centre of**

**46** **H. Percak**

**background curve and takes its full width at half the maximum value (further **
**called FWHM value) about ten times less than FWHM value of the background **
**curve.**

**In mean pressure range (from about 130 Pa to about 2000 Pa in the case **
**of C02-]Sr2-He laser) OPP curve shape results from inhomegeneous broadening, **
**caused by Doppler effect. This curve may be approximately described by Gaus**
**sian function, like the background curve in low pressure range.**

**P**

**Output power vs frequency curve of single-frequen**
**cy gas laser (OPP curve): ****v**** — frequency of laser **
**radiation, ****v0 — central frequency of OPF curve, ****P o M — output power of laser radiation, when **
**OPP curve is approximately described by the **
**Gaussian function, ****Pj) {v) — output power of laser ****radiation, when Lam b dip appears in the back**
**ground of approximately Gaussian OPP curve. **

Pod = Pd(vo)> Pog = Pq(vo)

**The FWHM values of OPF curve in mean and high pressure ranges are **
**similar. For example, in C02-N2-He laser these values vary from about 55 MHz **
**[6] to about 150 MHz [1]. For the same type of laser the FWHM value of **
**background curve in low pressure range is about 50 MHz, and the FWHM **
**value of Lamb dip is about 5 MHz [7].**

**The OPF curve in the case, which can be approximately described by Lo- **
**rentzian function, has an approximate analytical form**

**P i »** **0L**_{(v-v 0r + (AvJ2Y}**(^l/2)2** **(1)**

**where: PL(r) — output power of laser radiation when OPF curve is approxima**
**tely described by the Lorentzian function; v0 — central frequency of OPF ****curve; P0L ~ P L(r0); Avh — FWHM value of OPF curve (width at PL(v) **

**^ Pol/2)·**

**The OPF curve in the case, which can be approximately described by the **
**Gaussian function, has an approximate analytical form**

**where: PQ{v) — output power of laser radiation, when OPF curve is approxi****mately described by the Gaussian function; P0G c^PG(v0); Avlje — full width ****at 1 je maximum value (width at PG(v) ~ P 0G/e), e being the base of a natural ****logarithm.**

**In the case of function (2) there is a following interdependence between **
**FWHM value, Ava, and 1/e — width, Av1/e:**

**A****vg****~ (ln2)1/2zlr1/e ^ 0.832 Avl]e. ****(3)**

**In the process of laser frequency stabilization by means of automatic system, **
**an auxiliary modulation is used [8]. The frequency v of laser radiation is then ****described by the formula**

**v = v„ + Pmsin(27rvmi) ****(4)**

**where: ve — frequency of laser radiation without the auxiliary modulation; **

**Dm — frequency deviation of laser radiation caused by the auxiliary modula**

**tion ; vm — frequency of auxiliary modulation; t — time.**

**When the detunings (v — v0) of laser frequency v from the central frequency ****of OPF curve are little, having the form**

**I*— »„l s o . 1 ^ , ** **(5)**

**(where Avlj2 — FWHM value of OPF curve), the interdependence between ****laser output power P(r) and laser frequency v (the OPF curve) may be presented ****in an approximated form by the expansion in the Taylor series of P(v) function **
**in the neighbourhood of the central frequency v0 of OPF curve*. This expansion **
**in the Taylor series has a form**

**P(v) = P (v0)A****( V - V 0**_{2! }**)2 d*P(v) **_{dv*}

**in which is taken into consideration that**

**(**6**)**
**dnP(v)**

**dvn****(7)**

**when n is odd integer. Formula (7) follows from the symmetry of OPF function **

**P(v).**

*** The condition of little detunings (5) is determined in practice by the difference between ****the central frequency ****v0**** of OPF curve and the frequency of the nearest extremum of OPF **
**curve derivative. This practical condition of little detunings has a form**

**Iv-** **vqI** **(5a)**

**where the value of linearity coefficient jqin is about 0.58 in the case of Lorentzian OPF curve **
**or of Lamb dip and about 0.85 in the case of Gaussian OPF curve.**

**48** _{H. P}_{ercak}

**In the case of approximately Loren tzian OPF curve (formula (1)) we get**

**d2P M**

**dv2**_{(^}8 P,lOL)5

**In the case of approximately Gaussian OPF curve (formula (2)) we get**

**a*PQ(v)**

**dv2**_{(^,«/2>*}2PG(y o)**8P0Gln2 _{M»G)2}**

**(8)**

(**9**)

**It is possible to show that in the case of Lamb dip we get**

**d2PB(v) **

**dv2****=”0** **( ^8P(ODd)s**

**(1—2£d ln2)** (10)

**where: Pod**** — output power of laser radiation in Lamb dip centre v0; £D — **

**noefficient of Lamb dip narrowing.**

**The coefficient £D of Lamb dip narrowing is defined**

**£d**** = A****vg****/A****v****d** **(11)**

**where: A****vd**** — full width at the half depth value of Lamb dip; AvG — FWHM ****value of background curve PG(v) (see figure). This coefficient has in practice ****the following values:**

**£D ~ 5 - 1 0 . ** **(12)**

**When £d increases then Lamb dip contrast decreases, and vice versa.**

**The approximated form of functions PL(r), PG(r), PD(v) under little de**
**tunings condition (5) results from the formulae (6)-(10) and is expressed by the**
**respective relations:**
**Pt (v) ~ P 0L{1-(5x,/2)-*[(v-».)K]=}, ** **(13)**
**Pa(v) ^ P oa{ l-(3 Q/2)-2[(v -,0)MMn2}, ****(14)**
**PD(r)~P „D {l + (3D/2)-2[(v-r0)/»0P(l-2C 5!ln2)}, ** **(15)**
**in which**
**<5l**** = A****v****J****vq****, ****(16)**
**<3g**** = AvG/v0, ****(17)**
**<5jj = A****vjj****/****vq****. ****(18)**

**Substituting (4) to (13)-(15) we get a general formula**

**in which the coefficients A , B , G in Lorentzian curve, Gaussian curve and Lamb ****dip, are defined accordingly by the following formulae:**

**Pr = - X'L — ** **-“ -L t** **D,v, r. — vn**_{^0 ^0}**'m ye y0 ****5**
(**20**)
**(21)**
(**22**)
**(23)**
**B****g**** = - K Dm Ve~V0**G**V0 V0****5**
**r, ****, ^ An**
*B* ** D — + -« - D ---J**
**(24)**
**(25)**
**(26)**
**(27)**
**(28)**
**Coefficients Kh, KG, K B in formulae (23)-(28) are respectively defined by the ****following formulae:**
**8P**
**“ 4**
**8P0Lln2**
**5**2 ** _{G}** J

**(29)**

**(30)**

*** D = ^ ( 1 - 2 ^ 2 ) .**

_{°}**(31)**

**d****Synchronous detection applied in the process of laser frequency stabilization **
**by means of automatic system [8] to a signal of the frequency vm, yields the ****output voltage of synchronous detector. This voltage is proportional to the **
**value of coefficient PL or BG or PD (formulae (23)-(25)) in the case of frequency ****standards given in the approximate form of Lorentzian curve, Gaussian curve**

**50** **H. Percak**

or Lamb clip, respectively. This output voltage is independent of the values of
*coefficients AL, A G, AB (formulae (20)-(22)) and of CL, CG, <7D (formulae (26)— *
(28)). Therefore, this voltage is proportional to the relative detuning value
[(re — v0) /r0] of laser from the central frequency * v0 *of frequency standard, and to

the value of relative frequency deviation (Dm/r0) of laser subject to auxiliary
modulation. This conclusion is true when the condition (5), in which * v *is described

by formula (4), is satisfied.

One of the parameters characterizing the system of automatic frequency
stabilization of gas laser (further called AFSL system) is the so-called relative
threshold detuning [9], which may be also called a relative threshold error. It
is a minimal relative detuning * (\vs — v0\/v0) *corrected automatically. It is possible

to show that the value of relative threshold detuning (further called ETD value) is directly proportional to the value of coefficient Jl£1/2 in the case of

frequency standard given in the approximate form of Lorentzian curve, or to

*K G112, K B112 — in the case of frequency standards given respectively in the *

approximate form of Gaussian curve or of Lamb dip. Since the RTD value should be minimized, it is necessary to maximize the value of coefficients

*K G,K B (formulae (29)—(31)).*

The maximization of coefficients *, K G, KB can be obtained by maxi*
*mizing of the power P0L, P0G, P0D > respectively. In this case, however, there *

appear some restrictions which result from the fact, that the detector of laser
output power can not be charged with a too great radiation power. Therefore,
if the latter is too great, then before being delivered to power detector it must
be attenuated (using attenuating systems being on the outside of the laser) to
such a level, that its level value, averaging in period * (l/v m)* of modulating

signal, is not greater than continuous power limit of the applied detector. As
*a consequence, the new values of the coefficients AL, K G, K B are equal to the *
former ones divided by the value of the attenuation factor of the laser output power.

The reduction of the laser output power is necessary in the case of molecular lasers, because their output power can be many times greater than the conti nuous power limit of laser power detector. But in the general case of gas lasers the output power can be much smaller than the continuous power limit of laser power detector. Thus the laser output power should not be weakened but

maximized.

In accordance with the presented considerations, coefficients Pol > Pog> Pod

in formulae (29)-(31) in the case of molecular laser should be interpreted as the
values of laser output power (when generation frequency equals the central
frequency * v0 *of OPF curve) supplied to power detector in AFSL system, that

is the values of power attenuated to the level of continuous power limit of
laser power detector used in AFSL system. Therefore, in order to maximize
*the coefficients KG, K G, KB, in the AFSL system the laser power detector *
should have the possibly highest continuous power limit and be supplied with
laser output power attenuated to the value equal to the continuous power
limit of this detector.

*The maximization of coefficients KL, K G, K B may be also obtained by *
*respective minimization of the relative PWHM values <5L, dG, <3D (formulae *
(16)-(18)). However, the minimization possibilities of these values are very
limited, because the relative PWHM values <5L, dG, <5D of frequency standards are
determined by the features of gain medium. Due to an admissible reduction of
*this medium pressure, the values of parameters <5L, <5G, dB may be reduced at *
most two times with respect to their maximal values.

The three frequency standards (Lamb dip and approximately Lorentzian
curve and Gaussian curve) subject to analysis may be compared by comparing
the RTD values resulting from those standards. In accordance with the pre
*sented considerations we may apply such values of coefficients KL, K G, K B, *
which result from the typical values of coefficients <5L, <5g, <5d , and assume that
laser power detector used in AFSL system has for each analysed frequency
standard the same continuous power limit (Plim)max. The last assumption may be

written as

Pol — —Pod — (Piim)max· (32)

The analysed frequency standards may be compared by using the coefficients

**I K c \1/2**

*Qa ***l** = , (33)

**^DG = (pcT) ** **(34)**

*where: QGIj* — reduction coefficient of the RTD value of AFSL system in which

the maximum of approximately Gaussian OPF curve is used, as compared with
the corresponding value in this system in which the maximum of approximately
*Lorentzian OPF curve is used; QBG — defined like #GL for the centre of Lamb *
dip and the maximum of approximately Gaussian OPF curve.

After substituting (29)-(32) to (33) and (34) we get

«gl = -^ (ln 2 )·'2, (35)

In the case of molecular C02-ir2-He laser the typical values of parameters

*Avg and Avh are about 55 MHz [6] and about 100-150 MHz [1], respectively. *

From these values and from formulae (16) and (17) it follows

^l — (2 — 3) <5g. (37)

After substituting (37) to (35) and (12) to (36), we get the following evaluation
*of Q*gl* and QBG parameters *

*QGJj* -1 .7 -2 .5 ,

**52** **H. Percak**

**Analogically to (33) and (34) we may define coeffcient #DL (relating to the Lamb **
**dip and approximately Lorentzian OPF curve)**

**/ K \1'2**

= ( * ”-) · <40>

**From formulae (33), (34) and (40) we get the equation**

* Qjyii = Qd gQgL·· * (

**4 1**)

**This equation and the evaluations (38) and (39) yield the evaluation of coeffi**
**cient**

**$dl —10 30. ** **(42)**

**From (38) it follows, that the maximum of approximately Gaussian OPF **
**curve gives the RTD value of AFSL system about two times smaller than that **
**for the maximum of approximately Lorentzian OPF curve. Therefore, the **
**frequency standard given by the maximum of approximately Gaussian OPF **
**curve is about two times better than that expressed by the maximum of appro**
**ximately Lorentzian OPF curve. Analogically, from the evaluations (39) and **
**(42) it results, that the centre of Lamb dip is frequency standard about ten **
**times better than the maximum of approximately Gaussian OPF curve and **
**about twenty times better than the maximum of approximately Lorentzian **
**OPF curve. Furthermore, as it follows from the formula (36), Lamb dip centre **
**is the better frequency standard the higher is the value of narrowing coefficient **

£d·_{Frequency standards curves apart from the possibly low relative FWHM }

**values required for minimizing the RTD values, should be characterized by **
**central frequency v0 of high constancy and reproducibility. The above requi****rements with respect to v0 result from the fact, that the condition**

**Av. = Av0 ****(43)**

**must be fulfilled, since then coefficients Bh, BG, BD (formulae (23)-(25)) remain ****constant. In (43) Ave and Av0 are the frequency increments of ve and v0, respecti****vely. Thus, and error signal in AFSL system (output voltage of synchronous **
**detector [8]) does not change when the condition (43) is fulfilled. Therefore, **
**AFSL system does not stabilize the changes of frequency v0 but follows it. Just ****for this reason high constancy and reproducibility of the central frequency v0 ****of frequency standard are required.**

**The value of standard frequency v0 inconstancy in the case of Lamb dip ****centre or of approximately Gaussian or Lorentzian OPF curves maximum is **
**of the same order of magnitude. This value changes due to the changes of the **
**following gas gain mixture parameters:**

**i) temperature,**

**ii) composition and total pressure, and**
**iii) electric discharge conditions.**

**Therefore, the constancy and the reproducibility of standard frequency v0 ****may increase when the following parameters of gas gain mixture are stabilized:**

**1. Temperature (for example, by stabilization of water temperature cooling **
**a laser discharge tube).**

**2. Composition and total pressure.**

**3. Intensity of current flowing through the laser discharge tube. **

**References**

**[1] Sasnett** **M. W ., Reynolds** **R. S., IE E E J. Quant. Electron. Q E -7 (1971), 372.**
**[2] Bordź** **C., Henry L ., IE E E J. Quant. Electron. Q E -4 (1968), 874.**

**[3] Kęcki Z ., ***Podstawy spektroskopii molekularnej* **(in Polish), P W N , Warszawa 1975, p. 21.**
**[4] Maitland** **A ., Dunn** **M. H ., ***Laser Physics,* **North-Holland Publ. Co., Amsterdam 1969.**
**[5] Leś** **Z ., ***Wstęp do spektroskopii atomowej* **(in Polish), P W N , Warszawa 1972.**

**[6] Truefert** **A ., Vautier** **P.-, Onde Electr. 46 (1966), 417.**

**[7] Bourdet** **T ., OrszagA ., Valence** **Y ., C. R. Acad. Sc. (Paris) 277 (1973), B207.**
**[8] Percak** **H ., ***Radiation-frequency stabilization o f molecular lasers* **(in Polish), Scientific **

**Papers of the Institute of Telecommunication and Acoustics of the Technical University **
**of Wroclaw No. 30, Ser. Monographs No. 12, Wroclaw 1978.**

**[9] PercakH ., [in] ***Abstracts·. E K O N -7 8 , VIII Conference on Quantum Electronics and N on*

*linear Optics,*

**[Ed.] Institute of Physics, Adam Mickiewicz University, Poznań 1978.**

**Sect. A , p. 41.**

*Received June 21, 1985 *
*in revised form October 8, 1985*
A najiH 3 cbohctb KBauTOBbix cxaiuiapTOB q acro T b i b BHge MaKCHMyMa

KpilBOH 3aBKCHMOCTH BbIXOgHOH MOU|HOCTH OT HaCTOTbl OflHOHaCTOTKOrO
ra 3 0 B o ro Jia3epa, **a TaKJKe **b BHjje u e m p a n p o a a jia **Jl3M6a**

ITpOBOgHTCH OpHTHHEJIbHOe TeOpeTHHeCKOe o6bHCHeHHe KOHCTaTHpOBaHHBIX g o CHX n o p TOJIbKO 3KCne-
pHM eHTajibHo pa3JiHHuft b CBoiiCTBax (JjyHgaM eHTajibHtix KBaHTOBbix cTaHgapTOB nacTOTbi. **B **p a6 o T e

gOKa3aHO, HTO MaKCHMyM KpHBOft 3aBHCHMOCTH BblXOgHOH MOIgHOCTH OT TaCTOTbl OgHOHaCTOTHOrO **ra- **

**30**B oro Jia3 epa (Ha3BaHHoe g ajib m e MaxcHMyM K puBoii **BMOJI) **hbuhctch CTaHgapTOM nacTOTbi npuG im -

3HTejibHo b gB a p a3 a jiynm uM b cjiy n ae KpHBofi **BMOJI, **onucaHHOH (JjyHKUHeft T ay c ca , neM 4>yHKgneft

JIo p eH g a. H oK a3aH o Toace, mto geH Tp npO B ana Jl3M 6a HBjiseTca CTaHgapTOM nacTOTbi npH6jiH3HTejibHO

gecflTb p a3 jiyHuiHM, neM MaKCHMyM k ph b o h **BMOJI, **onucaHHOH (JjyHKimeii T ay cca, **a **T arace npH 6m i3H -